The Role of the Galaxy in the Dynamical Evolution of Transneptunian Objects

Home , Galactic tide, Yerkes Observatory

Duncan et al.: Galactic Perturbations of Minor Bodies 315

Martin J. Duncan Queen’s University at Kingston, Canada

Ramon Brasser Queen’s University at Kingston and Canadian Institute for Theoretical Astrophysics

Luke Dones and Harold F. Levison Southwest Research Institute

Our understanding of the past and present dynamical imprint of the galaxy on the outer reaches of our solar system has evolved considerably since the pioneering work of Oort (1950). In particular, the recent discoveries of objects with perihelia well beyond Neptune (Gladman et al., 2002; Brown et al., 2004) may be our first in situ glimpse of members of the hitherto hypothetical inner Oort cloud. This chapter reviews our current understanding of the formation and evolution of the Oort cloud and summarizes aspects of the problem needing to be explored in the near future.

1. INTRODUCTION dial solar nebula. Finally, Section 8 discusses future avenues of exploration, both observational and theoretical, that may Much of the current interest in the transneptunian region help us further understand the formation and evolution of arises from the possibility that the cometary reservoirs the the cometary reservoirs in the transneptunian region. therein have preserved some record of the formational proc- esses that led to the origin of our solar system. In this chap- 2. CONCEPTS OF THE OORT CLOUD ter we begin by describing what are thought to be the key ingredients in understanding the interplay between galac- Before proceeding to a detailed review of current mod- tic and solar system perturbations on minor bodies in the els of OC formation and evolution, we will briefly review outer solar system that may have led to the formation and key concepts concerning its structure. Details of the orbital dynamical evolution of the Oort cloud (henceforth referred characteristics of long-period comets (those with periods to as the OC).We then focus on the dramatic observational longer than 200 years, hereafter called LPCs) and of the discoveries that have prompted a reexamination of our mod- early history of this field are summarized in several exten- els and describe recent efforts to include some of the proc- sive reviews (e.g., Bailey et al., 1990; Fernández and Ip, esses that were important during the solar system’s forma- 1991; Weissman, 1990, 1996, 1999; Festou et al., 1993a,b; tive years. Wiegert and Tremaine, 1999; Dones et al., 2004 — hereaf- The chapter is organized as follows. In section 2 we ter DWLD — and other papers in the Comets II book). We briefly review the history of key concepts associated with do not describe studies of the formation of individual trans- the OC. Section 3 discusses the major perturbations on com- neptunian objects (TNOs); this topic is discussed by, e.g., etary orbits and summarizes the key timescales associated Greenberg et al. (1984) and Weidenschilling (1997, 2004), with each. Section 4 discusses how one can estimate the and reviewed in the chapter by Kenyon et al. mass of the outer cloud and section 5 discusses clues about In his historic paper, Oort (1950) proposed that the Sun the structure interior to the outer OC provided by observa- was surrounded by a spherical cloud of comets with semi- tions of Halley-type comets. In section 6, we present the major axes of tens of thousands of AU. In Oort’s model, results of an idealized model of OC formation called the planetary perturbations (primarily by Jupiter) acting on small “reference model,” in which the outer planets are assumed bodies placed the comets onto large, highly eccentric or- to have their current masses and orbits, the protoplanetary bits, after which perturbations by passing stars raised the gas nebula has dissipated and the Sun’s galactic environ- comets’ perihelia from the planetary region. Oort showed ment is assumed to be its current one. Section 7 describes that comets in the cloud are so far from the Sun that per- the results of recent simulations that model OC formation turbations from random passing stars can change their or- in cases in which the Sun formed in an embedded star clus- bital angular momenta significantly and occasionally send ter, and in some cases including the effects of the primor- some comets back into the planetary system as potential

315 316 The Solar System Beyond Neptune

LPCs. [Although Öpik (1932) first showed that stellar per- et al., 1987; Heisler et al., 1987). These periodicities have turbations could raise the perihelia of minor bodies being not held up under scrutiny (Jetsu and Pelt, 2000; but cf. scattered by the giant planets, he specifically rejected the Muller, 2002). However, despite the lack of direct evidence idea that comets in the cloud could ever be observed, even for the inner OC, its existence is still generally accepted. indirectly, because he did not recognize that stellar perturba- Some authors (e.g., Bailey et al., 1990) have stated that the tions would also cause some orbits to diffuse back into the inner OC must exist in order to replenish the outer OC, planetary region.] The relative importance of the four giant which might be completely stripped by passages of molecu- planets in populating the OC was later studied by many au- lar clouds. [But cf. Hut and Tremaine (1985); we return to thors (e.g., Safronov, 1972; Duncan et al., 1987, henceforth this claim in the next section.] DQT87). Indeed, two members of the inner OC’s population may Oort already recognized that the energy distribution of recently have been found: the unusual body (90377) Sedna observed LPCs did not agree with his model. Specifically, (a = 501 AU, q = 76 AU) (Brown et al., 2004) may be a there were too few “returning” comets with semimajor axes member of the inner OC as may be the object 2000 CR105 a < 104 AU relative to the number of “new” comets with (a = 224 AU, q = 44 AU) (Gladman et al., 2002). Whether a > 104 AU. Oort therefore had to invoke an empirical “fad- the unusually large perihelia of these objects was produced ing” law to hide many of the comets that had recently by a passing star (Morbidelli and Levison, 2004; Kenyon passed through the planetary region. Cometary fading has and Bromley, 2004) is a matter of some debate since other been modeled by Whipple (1962), Weissman (1980), Bailey models for their origin exist (Matese et al., 2005; Gomes (1984), Wiegert and Tremaine (1999), and Levison et al. et al., 2006; Gladman and Chan, 2006). However, if Sedna (2001, 2002), while Jewitt (2004) reviewed physical loss in particular is representative of the inner regions of the mechanisms for comets. No one has found a dynamical OC, then the inner OC may be rather massive [containing mechanism that resolves the fading problem (Wiegert and as much as 5 M in the rough estimation of Brown et al. Tremaine, 1999). The only known way to make models (2004)]. In section 7 we shall explore models of OC forma- agree with the observed orbital distribution is to allow the tion that produce a massive inner cloud as a byproduct of comets to become extinct, i.e., they must stop their produc- the Sun’s presumed early history in the denser environment tion of gas and dust. A comet can become extinct if a mantle of an embedded star cluster. But first we review the key fea- forms on its surface, thereby shutting off outgassing (Brin tures of the dynamical evolution of comets in the current and Mendis, 1979), or if the nucleus breaks into many small solar system environment. pieces (Levison et al., 2002). Catastrophic disruption was observed, for example, for Comet C/1999 S4 (LINEAR) 3. PERTURBATIONS ON COMETARY ORBITS (Weaver et al., 2001, and other papers in the May 18, 2001, issue of Science). However, Neslusan (2007) has recently The process by which small bodies evolve from the plan- argued that an underestimation of cometary fading in pre- etary region involves several stages during which the body’s vious studies has led to an overestimation of the flux of new path evolves under the gravitational effects of the Sun and comets by as much as a factor of 10. This, in turn, may help the following potential perturbers. to alleviate the problem of what has previously been thought to be the prediction of many more old “dead” comets than 3.1. Planets are observed. Therefore, the relative importance of extinction vs. fragmentation and disruption remains an issue that Assuming that comets formed in the region of the giant is under investigation. planets, their orbits initially evolve due to gravitational scat- Hills (1981) showed that the apparent inner edge of the tering by the planets. At first, planetary perturbations pro- ≈ 4 OC at a semimajor axis a = aI (1–2) × 10 AU could be a duce comparable changes in the comets’ semimajor axes (a) selection effect due to the rarity of close stellar passages and perihelion distances (q) when cometary eccentricities capable of perturbing comets with a < aI onto orbits with (e) are small. Eventually, most comets are placed onto highly perihelion distances small enough to make them observable. eccentric orbits with perihelia still in the planetary region. Most comets (and perhaps the great majority of comets) The planets continue to change the comets’ orbital energies might reside in the unseen inner OC at semimajor axes of (i.e., a) via a random walk (Yabushita, 1980), while leaving a few thousand AU. However, during rare close stellar pas- their angular momenta (or, equivalently, q for highly eccen- sages, “comet showers” could result (Hills, 1981; Heisler tric orbits) nearly unchanged. et al., 1987; Fernández, 1992; Dybczynski, 2002a,b). Comet showers caused by various astronomical mechanisms, such 3.2. Stars as passages of the Sun’s hypothetical companion star, Nem- esis, through the inner OC, were invoked to explain claimed Stars with masses above the hydrogen-burning limit of periodicities in crater formation and mass extinction events 0.07 M pass within 1 pc (~2 × 105 AU) of the Sun about on Earth (Shoemaker and Wolfe, 1986, and other papers in once per 105 yr (Garcia-Sánchez et al., 1999, 2001). The The Galaxy and the Solar System; Hut et al., 1987; Bailey average mass of these stars is ~0.5 M (Chabrier, 2001). Duncan et al.: Galactic Perturbations of Minor Bodies 317

Brown dwarfs (stellar objects with masses between 0.01 and modulates cometary perihelion distances and must be con- 0.07 M ) are about as numerous as stars in the solar neigh- sidered in studies of the influx of LPCs and Halley-type borhood, but their average mass is only ~0.05 M (Chabrier, comets (Matese and Whitmire, 1996; Levison et al., 2006). 2002, 2003). Thus brown dwarfs probably do not perturb the OC much. In the impulse approximation, the change in 3.4. Molecular Clouds velocity Δv of an OC comet with respect to the Sun due to a stellar passage is given by While stars and tides have been the only perturbers included in most models of the OC, it has been suggested that Δ 2GM* ˆ ˆ v = (dC/dC – dS/dS) molecular clouds (MCs) (see Williams et al., 2000, for a V* review) might be very destructive of the outer OC (Bier-

where G is the gravitational constant, M* is the mass of the mann, 1978; Napier and Clube, 1979; Bailey, 1983, 1986; star, V* is its speed relative to the Sun, and dC, dS are the Hut and Tremaine, 1985; Weinberg et al., 1987; Bailey et impact parameters for the encounter with respect to the al., 1990). Napier and Staniucha (1982) claimed that giant comet and the Sun, respectively (Oort,1950). The strongest molecular clouds (GMCs) have removed ~99.9% of the encounters are those that have dS << dC or dC << dS, i.e., those comets from the outer OC. Hut and Tremaine (1985) found in which the star perturbs either the Sun or the comet much that MCs were about as effective as stars in stripping the more than it does the other object. In fractional terms, stars OC; for either type of perturber, acting alone, the half-life change q much more than they change a. This is a conse- of a comet at a = 25,000 AU (0.12 pc) is about 3 G.y., im- quence of the long lever arm and slow speed of comets on plying a net half-life of 1.5 G.y. The half-life is shorter at highly eccentric orbits near aphelion [see Appendix B of larger semimajor axes. Weinberg et al. (1987) found slightly Eggers (1999) for an elaboration of this argument]. How- shorter timescales. At face value, these results imply that ever, as we discuss next, tides from the galactic disk are most of the outer OC is lost, so that perhaps only 10% of slightly more effective than stars in producing systematic the comets survive for 4 G.y. However, the local mass den- changes in q. Nonetheless, passing stars do produce a ran- sity in molecular clouds and other parameters for clouds dom walk, or “diffusive” change, in cometary semimajor remain uncertain. Hut and Tremaine (1985) argue that the axes (Weinberg et al., 1987). existence of wide binary stars with separations on the order of 20,000 AU implies that molecular clouds cannot be 3.3. The Galactic Tidal Field vastly more destructive to the OC than stars. Thus we will neglect molecular clouds in what follows. The importance of galactic tides, i.e., the differential gravitational acceleration of OC comets relative to the Sun 3.5. Other Galactic Perturbers due to the disk and bulge of the Milky Way, was pointed out by Byl (1983, 1986, 1990), Smoluchowski and Torbett Giant molecular clouds were actually postulated by (1984), Heisler and Tremaine (1986), Delsemme (1987), Spitzer and Schwarzschild (1953) before molecular clouds and Matese et al. (1995). Slightly less than half of the lo- were discovered. Spitzer and Schwarzschild were trying to cal galactic mass density is in stars. Thus at most times (i.e., explain “disk heating,” i.e., the observation that older stars at times other than during a strong comet shower), the rate within the galactic disk typically have larger random veloci- at which comets are fed into the planetary region from the ties than younger stars. In their model, gravitational scat- OC due to the galactic tide is probably slightly larger than tering of stars by GMCs progressively excites stellar veloci- the influx due to stellar passages (Heisler and Tremaine, ties over time. However, it appears that additional sources 1986). Indeed, Heisler (1990) concluded that when stellar of disk heating are required, since GMCs appear incapable impulses are added to the galactic tidal interaction, the long- of producing the velocity dispersion of the dynamically hot- timescale average increase in the steady state flux due to the test disk stars, such as white dwarfs (Binney and Tremaine, tide alone is only ~20%. Tides change comets’ q at nearly 1987). Thus additional perturbers of stars, and potentially constant a. The disk (“z”) component of the galactic tide the OC, such as spiral arms (Jenkins and Binney, 1990) and causes q to oscillate in and out of the planetary region with a possibly massive black holes in the galactic halo (Hänninen period Tz that is on the order of 1 G.y. for comets with a ~ and Flynn, 2002), have been invoked. We ignore spiral arms 10,000 AU and initial q ~ 25 AU (equation (18) in Heisler and galactic perturbers other than the smooth component of –3/2 and Tremaine, 1986; DQT87). The value of Tz scales as a , the tide and passing stars in this paper. i.e., inversely as the comet’s orbital period. In addition, there is a “radial” component of the galactic tide due to the mass 3.6. Other Perturbers Within the Solar System interior to the Sun’s orbit around the galaxy. The amplitude of the radial tide is a factor of ~8 smaller than the disk tide, There have been suggestions that the OC may contain a but the radial tide is still important because it breaks conser- red dwarf (Davis et al., 1984; Whitmire and Jackson, 1984), vation of Jz, the component of a comet’s angular momen- a brown dwarf or giant planet (Matese et al., 1999; Murray, tum perpendicular to the galactic plane. Thus the radial tide 1999; Horner and Evans, 2002), or an Earth-mass planet 318 The Solar System Beyond Neptune

Fig. 1. Key timescales relevant to the dynamical evolution of a comet in the current solar system are plotted against semimajor axis.

These include (1) orbital period (dotted line), (2) the energy diffusion time td (timescale for cumulative planetary perturbations near pericenter to change the comet’s semimajor axis by roughly a factor of 2) for several pericentric distances q (solid lines), and (3) the tidal torquing time tq (timescale for the galactic tidal field to change q by roughly 10 AU) for the current galactic disk density (dashed line).

(Goldreich et al., 2004). Wiegert and Tremaine (1999) in- next passage through the planetary region; if the other case cluded solar companions and massive circumsolar disks in occurs, i.e., the line of tq is crossed before the period line, some of their simulations. While the existence of extra per- the comet’s perihelion is often lifted out of the planetary turbers cannot be disproved, we will not include them in region by the tide and stars quickly enough to be saved from this review, in the absence of compelling evidence for their being ejected. The intersection of the lines of td and tq in- reality (see the chapter by Gomes et al.). dicates the value of a at which the lifting is likely to begin and is a measure of the inner edge of the OC. 3.7. Key Timescales Thus, Fig. 1 shows that comets with pericenters in the Jupiter-Saturn zone (q ~ 5–15 AU) will diffuse on timescales 4 7 As discussed above, to a good first approximation the of 10 –10 yr to the point where the energy diffusion time td main perturbers in the current solar system are the four giant is comparable to the orbital period P (a ~ several hundred planets and the z-component of the galactic tide. The time- to 10,000 AU, depending on q), at which point the diffu- scales shown in Fig. 1 (adapted from DQT87) can then be sion approximation breaks down since the energy kick in used to follow the expected evolutionary path of cometary one orbit is comparable to the binding energy of the orbit. orbits once the comets are scattered onto eccentric orbits Most comets in this range of q will subsequently be ejected, with a >> q. Such comets tend to diffuse outward in semima- although due to the wide range in possible planetary “kicks” jor axis along lines of constant q, so that they evolve ap- in energy, a small fraction (typically a few percent) can be proximately along the lines of td, which are slowly decreas- launched to the region where the time for the pericenter to ing with increasing a. Two things can happen to the comet, be lifted away from the Jupiter-Saturn region is less than depending on which line it crosses first: If it crosses the an orbital period (a > 20,000 AU, as can be seen in Fig. 1). period line (increasing with a) before it crosses the line of On the other hand, comets with pericenters in the Uranus- 8 tq, the comet has a high probability of being ejected on the Neptune zone (q ~ 25 AU) will diffuse on 10 -yr timescales Duncan et al.: Galactic Perturbations of Minor Bodies 319

until they reach the region where the tidal torquing time is total brightness (generally dominated by coma) to cometary comparable to the period (a < 10,000 AU from Fig. 1). In masses. In Weissman’s calibration, which is based upon 1P/ the next orbit, depending on the argument of perihelion with Halley’s measured size and total brightness and an assumed respect to the galactic plane, roughly half the comets will be density of 0.6 g cm–3 [cf. the determination of a density of torqued outward to the relative safety of the OC and roughly 0.6 (+0.5, –0.3) g cm–3 for Comet 9P/Tempel 1 by A’Hearn half will be drawn into the Jupiter-Saturn zone, where they et al. (2005)], the diameter of a comet with H10 = 11 is d11 = are likely to be ejected. Similarly, objects with q ~ 35 AU — 2.3 km, and the mass of a comet with H10 = 11 is m11 = i.e., those in the scattered disk — will evolve on billion-year 4 × 1015 g. Weissman (1996) assumes, based on Everhart timescales to a ~ 3000 AU where their perihelia will be (1967), that LPCs have a shallow size distribution (with in- torqued in or out. dex 2 for the cumulative distribution) up to a size dcrit ~ Figure 1 can also be used to understand the key ob- 20 km, and follow a steep distribution at larger sizes. For servational bias concerning the OC noted by Hills (1981). these assumptions, most of the mass in the OC is in bodies Observed LPCs come directly from the “outer” OC (a > with diameters between d and d , and the average mass – 11 crit 20,000 AU), where the galactic tidal field can drive the peri- of a comet M is on the order of dcrit/d11 × m11, or m ~ 4 × 16 12 helion distance from q ~ 15 AU to q < 3 AU (where they 10 g. Weissman assumes NO = 1 × 10 , giving a mass for 28 produce a visible coma) in one cometary orbital period. the outer OC of NOm = 4 × 10 g, i.e., 7 M . This estimate Comets with perihelia driven inward from the “inner” OC is extremely uncertain; since the size of a dynamically new (a < 20,000 AU) in the current galactic environment will LPC has never been directly measured, cometary densities do so over several orbits and thus at some time will pass may be even smaller than 0.6 g cm–3 (Davidsson and Gutiér- through perihelion when q is near Jupiter and/or Saturn and rez, 2004) and uncertainties in the extent of cometary fad- the planetary perturbations almost inevitably drive such ing may have lead to an overestimation of the flux of new comets back into interstellar space (or occasionally into the comets (Neslusan, 2007). outer OC) or into much more tightly bound orbits where Recently, Francis (2005) has used the data from the tidal torquing is negligible. Thus, Jupiter (and to a lesser LINEAR survey to provide revised estimates of the flux of extent Saturn) acts like a barrier preventing most objects LPCs, the dependence on perihelion distance and the ab- from the inner OC from directly being observed. This ef- solute magnitude distribution. He estimates that the outer fect is called the “Jupiter barrier.” Since inner OC comets OC contains ~5 × 1011 comets down to absolute magnitude 11 are rarely directly injected into visible orbits, the inner cloud H10 = 17 and ~2 × 10 comets down to absolute magni- could be quite massive, as we discussed in section 2. tude H10 = 11. The latter estimate is 5× lower than that used Note that since the tidal torquing time is inversely pro- by Weissman (1996), which would predict an outer OC mass portional to the local galactic density, the “lifting” of com- of 1.4 M if all of the other parameters of Weissman’s fit re- etary perihelia and hence the inner edge of the OC will oc- mained unchanged. Like Weissman, Francis (2005) assumes cur at smaller radii for larger densities. We shall return to a two-component magnitude distribution, with a break at this issue is section 7. magnitude Hb = 6 or 6.5 (cf. Everhart, 1967; Hughes, 2001). The magnitude distribution for “faint” comets (i.e., comets 4. THE MASS OF THE OORT CLOUD with H10 > Hb) that Francis (2005) finds is shallower (more top-heavy) than Weissman (1996) assumes. However, the The present-day mass of the OC may provide interesting mass of the outer cloud is likely dominated by “bright” constraints on models of planet formation, under the as- comets (H10 < 6–6.5), which are poorly constrained by the sumption that comets represent planetesimals from the re- LINEAR data. gion of the giant planets. If the OC’s mass can be estimated We find that the mass of the outer OC could be between from observations and the efficiency of OC formation can ~1 and 60 ME, depending upon the assumed magnitude dis- be determined by dynamical models, we can hope to infer tribution for bright comets (Everhart, 1967; Hughes, 2001) the total mass in the planetesimal disk during the epoch and the relationship between absolute magnitude and com- when the OC formed. The formation timescale and migra- etary mass (Weissman, 1996; Bailey and Stagg, 1988). We tion histories of the giant planets, in turn, depend critically obtain a low value for the outer cloud’s mass if we assume on the mass of the planetesimal disk (Hahn and Malhotra, that it contains very few comets with H10 < 3, the magnitude 1999; Gomes et al., 2004). of the brightest comet seen by LINEAR thus far (Francis, From the observed flux of new comets, Heisler (1990) 2005). This assumption yields an outer cloud mass of 3 ME used her model-determined cometary influx rate, and the using Weissman’s (1996) scaling or 0.6 ME using Bailey and assumption that the current flux equals the long-term aver- Stagg’s (1988). However, an abrupt cutoff for any value of age, to infer that the present-day outer OC contains NO = H10 < 0 seems unlikely, since Hale-Bopp, which had H10 = 5 × 1011 comets with a > 20,000 AU. This estimate refers –0.8 (Andreas Kammerer, kometen.fg-vds.de/koj_1997/ to comets with “absolute magnitude” H10 < 11. Weissman c1995o1/95o1eaus.htm) probably originated in the OC. (1996) reviews estimates of the number of comets in the OC Given the very poorly calibrated relationship between H10 and attempts to relate H10, which is a measure of a comet’s and cometary mass, we cannot place a lower (or upper) limit 320 The Solar System Beyond Neptune on the mass of the outer cloud from the observed rate of Recall that the other, more numerous class of short-period cometary passages through the inner solar system. Low comets are the Jupiter-family comets (hereafter JFCs; short- outer cloud masses, on the order of 1 M , are more consis- period comets with Tisserand parameters with respect to Ju- tent with current models of the dynamical evolution of the piter, T, greater than 2) (Carusi et al., 1987; Levison, 1996). outer solar system (see section 6). Most of the JFCs are believed to leak in from the inner edge of the scattered disk (Duncan and Levison, 1997; Duncan et 5. CLUES FROM THE CAPTURE OF al., 2004), initially under the dynamical control of Neptune. HALLEY-TYPE COMETS As might be expected, the integrations of LDD01 showed that low-inclination objects with initial semimajor axes in Due to the “Jupiter barrier” (or more accurately the “Ju- the inner OC also evolve into JFCs if their perihelia origi- piter-Saturn barrier”) discussed in section 3.7, bodies from nated near Neptune. However, those low-inclination comets the inner OC for which the galactic tide is driving their peri- from the inner OC with initial q < 25 AU generally evolved helion distances into the region of the giant planets will suf- into HTCs. Thus, in order for a flattened inner OC to be the fer perturbations that will inevitably drive such comets back source of the low-inclination HTCs, inner OC comets must into the outer OC (or interstellar space) or into much more be able to evolve onto orbits with q < 25 AU and remain tightly bound orbits where tidal torquing is negligible. A on low-inclination orbits in the inner OC for the age of the small fraction of the latter comets will eventually diffuse solar system. down into orbits with sufficiently small semimajor axes that LDDG06 showed that (1) only if a < 3000 AU are ga- their perihelia will begin to be substantially affected and lactic tides weak enough that orbital inclinations do not they can then come sufficiently close to the Sun to produce change significantly over the age of the solar system (i.e., comas. Numerical simulations (Duncan et al., 1988; Quinn the OC remains disk-like), and (2) only if a > 12,000 AU et al., 1990; Emel’yanenko and Bailey, 1998) have sug- are galactic tides strong enough that an object can evolve gested that these objects may be an important source of from an orbit with q > 30 AU to one with q < 25 AU before Halley-type comets (short-period comets — periods less perturbations from Neptune remove it from the OC. This than 200 yr — with Tisserand parameters with respect to obvious inconsistency is resolved if the flattened source is Jupiter, T, less than 2) (Carusi et al., 1987; Levison, 1996). itself being dynamically replenished, rather than being a Thus, it has been suggested that Halley-type comets (here- fossilized remnant of the early history of the solar system. after HTCs) may represent our only currently observable Thus, LDDG06 studied the dynamical evolution of objects link to the inner OC. that evolve off the outer edge of the scattered disk and have Levison et al. (2001, henceforth LDD01) attempted to their perihelion distances driven inward by galactic tides. constrain the structure of the inner OC by modeling the They found that roughly 0.01% of these objects evolve onto “capture” process by which OC comets evolve onto orbits HTC-like orbits. The orbital element distribution of the re- like those of HTCs. While some HTCs, such as Halley and sulting HTCs is consistent with observations, including the Swift-Tuttle, follow retrograde orbits, most (19 of 26 cur- requisite number of retrograde orbits, that are produced by rently known with q < 1.3 AU — see www.physics.ucf.edu/ the tidally driven precession of the line of nodes in galac- ~yfernandez/cometlist.html) revolve on prograde orbits. tic coordinates. LDD01 found that cometary inclinations were roughly con- In order for the scattered disk to supply enough HTCs, served during the capture process, so that the source region the model predicts that it needs to contain 3 billion comets had to be somewhat flattened. [An alternative, proposed by with diameters larger than 10 km. This value is larger than Fernández and Gallardo (1994), is that the observed pre- estimates inferred from observations of the scattered disk ponderance of prograde orbits is due to cometary fading: and it may be larger than that needed for the JFCs. How- When the number of perihelion passages is limited by phys- ever, the structure of the scattered disk at large heliocen- ical causes, fewer retrograde comets are found in evolved tric distances, where the HTCs would come from, is not states since they evolve dynamically more slowly than their constrained by either the JFC models or observations. In prograde counterparts. However, LDD01 modeled this possi- addition, LDDG06’s HTC model lacked passing stars and bility in their detailed numerical integrations and found that molecular clouds, which could affect the delivery rates. for the range of fading times proposed by Fernández and It should be noted that, in the simulations of LDDG06, Gallardo, the simulated HTCs generally had larger semima- comets coming off the outer edge of the scattered disk not jor axes than are observed.] Since the outer OC is roughly only become HTCs, but contribute to the LPC population, spherical, LDD01 concluded that most HTCs must derive especially to those LPCs known as dynamically new com- from a flattened inner core which they associated with the ets (DNCs) [a > 10,000 AU; see Levison (1996), although inner OC. LDD01 found that models in which the median Dybczynski (2006) argues that a > 25,000 AU is a better cri- inclination of inner OC comets, i', is between 10° and 50° terion]. In particular, the models produce a population of can fit the HTC orbital distribution. DNCs with a median inclination of 40° and in which only However, Levison et al. (2006, henceforth LDDG06) 30% are retrograde. The observed inclination distribution subsequently realized that there was a problem in identify- of the DNCs is isotropic. However, one interesting aspect ing the flattened outer source with a “fossilized” inner OC. of the model DNCs is that they have large semimajor axes: Duncan et al.: Galactic Perturbations of Minor Bodies 321

75% of DNCs in LDDG06 have a > 30,000 AU. This is with ~90% of those comets in orbits with semimajor axes relevant because Fernández (2002) performed an analysis between 500 and 20,000 AU. of the inclination distribution of LPCs in different ranges The first study using direct numerical integrations to of semimajor axes and concluded that although the DNCs model the formation of the OC was that of DQT87. To save are isotropic, 68% of the DNCs with a > 32,000 AU are pro- computing time, DQT87 began their simulations with com- grade. This is consistent with LDDG06’s model if most of ets on low-inclination, but highly eccentric, orbits with peri- these objects are from the scattered disk. To test the latter helia in the region of the giant planets. Gravitational pertur- requirement, future investigations combining the simula- bations due to the giant planets and the disk (z) component tions just discussed together with a complete evolutionary of the galactic tide were included. A Monte Carlo scheme model of the OC will be needed. from Heisler et al. (1987) was used to simulate the effects of stellar encounters. 6. SIMULATIONS OF OORT CLOUD The main prediction of DQT87 was that the OC is cen- FORMATION: THE REFERENCE MODEL trally condensed, with roughly 4–5× as many comets in the inner OC (a < 20,000 AU) as in the classical outer OC. In In his 1950 paper, Oort did not consider the formation their model, comets with q0 > 15 AU are much more likely of the comet cloud in detail, but speculated that the comets to reach the OC and survive for billions of years than are were scattered from the asteroid belt due to planetary per- comets with smaller initial perihelia. For example, only 2% turbations and lifted by stellar perturbations into a large of the comets with q0 = 5 AU should occupy the OC at cloud surrounding the solar system. Oort proposed the as- present, while 24% of the comets with q0 = 15 AU and 41% teroid belt as the source region for the LPCs on the grounds with q0 = 35 AU should do so. This result appeared to con- that (1) asteroids and cometary nuclei are fundamentally firm that Neptune and Uranus, which have semimajor axes similar in nature and (2) the asteroid belt was the only stable of 30 and 19 AU, respectively, are primarily responsible for reservoir of small bodies in the planetary region known at placing comets in the OC. However, this finding is ques- that time. Kuiper (1951) was the first to propose that the tionable, since the highly eccentric starting orbits had the icy nature of comets required that they be from a more dis- consequence of pinning the perihelion distances of the com- tant part of the solar system, among the orbits of the giant ets at early stages. This, in turn, allowed Neptune and Ura- planets. Thus, ever since Oort and Kuiper’s work, the roles nus to populate the OC efficiently because they could not of the four giant planets in populating the comet cloud have lose objects to the control of Jupiter and Saturn. been debated. Kuiper (1951) proposed that Pluto, which was DWLD present results of a similar study to that of then thought to have a mass similar to that of Mars or the DQT87, but starting with “comets” with semimajor axes be- Earth, scattered comets that formed between 38 and 50 AU tween 4 and 40 AU and initially small eccentricities and in- (i.e., in the Kuiper belt!) onto Neptune-crossing orbits, after clinations. These initial conditions are more realistic than which Neptune, and to a lesser extent the other giant plan- the highly eccentric initial orbits assumed by DQT87. The ets, placed comets in the OC. study integrated the orbits of 3000 comets for times up to Later work (Whipple, 1962; Safronov, 1972) indicated 4 b.y. under the gravitational influence of the Sun, the four that Jupiter and Saturn tended to eject comets from the solar giant planets, the galaxy, and random passing stars. Their system, rather than placing them in the OC. The gentler model of the galaxy included both the “disk” and “radial” perturbations by Neptune and Uranus (if these planets were components of the galactic tide. The disk tide is propor- assumed to be fully formed) thus appeared to be more ef- tional to the local density of matter in the solar neighbor- fective in populating the cloud. However, their role was hood and exerts a force perpendicular to the galactic plane, unclear because the ice giants took a very long time to form while the radial tide exerts a force within the galactic plane in Safronov’s orderly accretion scenario. Fernández (1978) (see section 3.3). These simulations did not include other used a Monte Carlo, Öpik-type code and suggested that perturbers such as molecular clouds, a possible dense early “Neptune, and perhaps Uranus, could have supplied an im- environment if the Sun formed in a cluster (Gaidos, 1995; portant fraction of the total mass of the cometary cloud.” Fernández, 1997), or the effects of gas drag (de la Fuente Fernández (1980) extended this work by following the sub- Marcos and de la Fuente Marcos, 2002; Higuchi et al., sequent evolution of comets on plausible near-parabolic or- 2002). Studies involving the latter two effects are discussed bits for bodies that had formed in the Uranus-Neptune re- below. gion. He concluded that about 10% of the bodies scattered DWLD describe two sets of runs with dynamically “cold” by Uranus and Neptune would occupy the OC at present, and “warm” initial conditions. The results were very simi- and that the implied amount of mass scattered by the ice gi- lar, so they focus on the “cold” runs, which included 2000 ants was cosmogonically reasonable, i.e., not vastly greater particles with root-mean-square initial eccentricity, e0, and than the masses of Uranus and Neptune themselves. inclination to the invariable plane, i0, equal to 0.02 and 0.01 Shoemaker and Wolfe (1984) performed an Öpik-type radians, respectively. simulation to follow the ejection of Uranus-Neptune plan- DWLD take the results of these calculations at 4 G.y. to etesimals to the OC. They found that ~9% of the original refer to the present time. For a comet to be considered a population survived over the history of the solar system, member of the OC, they required that its perihelion distance 322 The Solar System Beyond Neptune exceeded 45 AU at some point in the calculation. For the see the chapter by Gomes et al.)] and then end up in the “cold” runs, the percentage of objects that were integrated inner OC. Figure 2 shows the time evolution of the popu- that currently occupy the classical “outer” OC (20,000 AU ≤ lations of the OC and scattered disk in the simulation. The a < 200,000 AU) is only 2.5%, about a factor of 3 smaller scattered disk is initially populated by comets scattered by than found by DQT87. The percentage of objects in the inner Jupiter and Saturn, and peaks in number at 10 m.y. (off- OC (2000 AU ≤ a < 20,000 AU) is 2.7%, almost an order scale on the plot). After this time the population of the scat- of magnitude smaller than calculated by DQT87. DQT87 tered disk declines with time t approximately as a power found a density profile n(r) ∝ r –γ with γ ~ 3.5 for 3000 AU < law, N(t) ∝ t–α, with α ~ 0.7. The predicted population of r < 50,000 AU, so that in their model most comets reside the scattered disk in this model at the present time is roughly in the (normally unobservable) inner OC. Fitting the entire 10% the population of the outer OC. OC at 4 G.y. in the models reported in DWLD to a single Figure 2 also shows the populations of the inner and power law yields γ ~ 3, shallower than the value found by outer OCs individually. The population of the outer OC DQT87. A value of γ ~ 3 implies that the inner and outer peaks around 600 m.y. while the inner OC peaks around OCs contain comparable numbers of comets at present in 1.8 G.y. Because of the faster decline of the outer OC, the this model. This result holds because most comets that begin ratio of numbers of inner to outer OC comets increases with in the Uranus-Neptune zone evolve inward and are ejected time, to 1.1 at present. As noted above, only 2.5% of the from the solar system by Jupiter or Saturn. A small fraction comets that were initially in the simulation occupy the outer of these are placed in the OC, most often by Saturn. How- OC at 4 G.y. ever, all four of the giant planets place comets in the OC, At face value, the low efficiency of OC formation in the albeit with different efficiencies. simulations reported in DWLD implies a massive primordial The OC is built in two distinct stages in the DWLD protoplanetary disk. Assuming an outer OC mass of 2– model. In the first few tens of million years, primarily the 40 M (Francis, 2005) (see section 4), the simulation effi- outer OC is built by Jupiter and Saturn; subsequently, most ciency implies that the original mass in planetesimals be- of the inner OC is built, mainly by Neptune and Uranus, tween 4 and 40 AU was ~80–1600 M , some 2–40× the with the population peaking about 800 m.y. after the be- mass in solids in a “minimum mass” solar nebula. The ginning of the simulation (Fig. 2). Objects that enter the OC amounts of mass at anything but the low end of this esti- during this second phase typically first spend time in the mate would likely have produced excessive migration of the “scattered disk” [45 AU ≤ a < 2000 AU, with perihelion giant planets and/or formation of additional giant planets distance <45 AU at all times (Duncan and Levison, 1997; (Hahn and Malhotra, 1999; Thommes et al., 2002; Gomes et al., 2004). However, the uncertainties discussed at the end of section 4 suggest that it is possible that the outer OC may currently contain ~1 M , in which case the disk mass inferred in DWLD does not present a problem. The results may be inconsistent with observations in an- other way. The population of the scattered disk that DWLD predicted, on the order of 10% the population of the OC, may be much larger than the actual population of the scattered disk inferred from observations of large bodies (Tru- jillo et al., 2000). Again, however, the large uncertainties in the observational estimates of both populations mean that there may be no problem. However, we must certainly come to terms with the fact that none of the models described above will produce the orbits of the inner OC objects described at the end of section 2. For that, we will likely need the types of models discussed next.

7. SIMULATIONS OF OORT CLOUD FORMATION IN A STAR CLUSTER

Tremaine (1993), Gaidos (1995), Fernández (1997), Eg- gers et al. (1997, 1998), Eggers (1999), and Fernández and Fig. 2. In the simulations reported in DWLD the outer OC, which is originally populated by comets injected by Jupiter and Saturn, Brunini (2000) have discussed star formation in different forms more rapidly than the inner OC. These simulations predict galactic environments. These authors point out that the Sun that the present populations of the inner and outer OCs should be may have formed in a denser environment than it now oc- comparable and that of the scattered disk should contain roughly cupies (i.e., in a molecular cloud or star cluster), and found 10% as many comets as the outer OC. that a more tightly bound OC would form. Duncan et al.: Galactic Perturbations of Minor Bodies 323

7.1. Simulations Without a Primordial Solar Nebula cluster, which are clusters that are very young and heavily obscured by dust since the molecular gas is still present A first attempt to simulate the formation of the OC when (Lada and Lada, 2003). Most stars in the galaxy probably the Sun is still in a star cluster was done by Eggers (1999). form in embedded clusters with between 100 and 1000 In that work, the formation of the OC was simulated for members (Adams et al., 2006); indeed, typical populations 20 m.y. using a Monte Carlo method with two star clusters, of embedded clusters within 2 kpc of the Sun today are 50– in which the stellar encounters occurred at constant time 1500 stars (Lada and Lada, 2003). The lifetime of the gas intervals and were computed analytically and the cluster in these clusters is typically 1–5 m.y.: Only 10% of em- tidal field was neglected. The first cluster had an effective bedded clusters last for 10 m.y. (Lada and Lada, 2003). density of 625 stars pc–3 and the other had an effective den- Since the formation of unbound stellar clusters is the rule sity of 6.25 stars pc–3, as compared with the current den- and not the exception (Lada et al., 1984), it is probable that sity of ~0.1 stars/pc3. Both clusters had a velocity disper- the Sun formed in such a cluster and escaped from it within sion of 1 km s–1. Eggers started with a population of comets <5 m.y. on nearly circular, low-eccentricity orbits in the region of Recently, Gutermuth et al. (2005) have observed three the giant planets, and defined a comet to be in the OC if it embedded clusters using near-IR data. The clusters in their evolved under the combined planetary and stellar pertur- sample were chosen because they are rich and relatively bations onto an orbit with q > 33 AU and a > 110 AU. With young. One of these clusters exhibits clumping of stars into these definitions, he obtained efficiencies of 1.7% and 4.8% three different regions. The peak volume densities in these for the loose and dense clusters respectively. Most objects clusters in gas and stars are on the order of 104–105 M pc–3, for the low-density cluster had a = 6–7 × 103 AU and the with mean volume densities ranging from 102–103 M pc–3. resulting OC was fairly isotropic. For the high-density clus- The highest peak volume density quoted is 3 × 105 M pc–3. ter, most objects were in the range a = 2–4 × 103 AU and In two of the clusters, their observations show that 72% and again had a fairly isotropic inclination distribution. 91% of the stars are in locations with stellar densities of It is worth noting that Eggers (1999) also used an N- 104 M pc–3 or larger, respectively. For the third cluster this body simulation to investigate the interesting proposal of fraction is 24%. These constraints on the stellar density were Zheng et al. (1990) that some fraction of OC comets might used by BDL06 to select a range in central densities for the have been captured from the intracluster medium during clusters in their simulations. star–star encounters. Eggers found that at most a few per- BDL06 adopted a model for the cluster that is often used cent of outer OC comets are likely to be of extrasolar ori- (e.g., Kroupa et al., 2001) and assume the ratio of stars to gin. Subsequently, motivated by the discovery of Sedna et gas is constant throughout the cluster. In this so-called al. (2004) and Morbidelli and Levison (2004), he indepen- Plummer model, the potential Φ(r) and density ρ(r) are given dently investigated the capture into the Sun’s inner OC of as a function of distance by (e.g., Binney and Tremaine, comets from the protoplanetary disks of other stars during 1987) very close stellar encounters. Fernández and Brunini (2000), henceforth FB2K, per- GM ρ formed simulations of the evolution of comets starting on Φ(r) = ; ρ(r) = 0 (1) 2 2 2 2 5/2 eccentric orbits (e ~ 0.9) with semimajor axes 100–300 AU (r + c ) (r + c ) and included an approximate model of the tidal field of the gas and passing stars from the cluster in their model. The where M is the total mass of gas and stars combined in the –3 ρ clusters had densities ranging from 10–100 stars pc , and cluster, r is the distance from the centre of the cluster, 0 is the density of the core of the molecular cloud in their the central density of the gas and stars combined, and c is models ranged from 500 to 5000 M pc–3. Their simulations the Plummer radius (which defines the length scale of the formed a dense inner OC with semimajor axes of a few hun- model; half the cluster mass is contained within a radius of dred to a few thousand AU. The outer edge of this cloud was 1.3c). BDL06 studied Plummer models with 200–400 stars, dependent on the density of gas and stars in the cluster. with a wide range of initial central densities. Since the simu- FB2K reported they were able to successfully save mate- lations spanned only the first few million years of the Sun’s rial scattered by Jupiter and particularly Saturn, which were lifetime, only the planets Jupiter and Saturn were included the main contributors to forming the inner OC, since Ura- and the planets were taken to have their current masses on nus and Neptune took too long to scatter material out to orbits appropriate to the period before the late heavy bom- large-enough distances (see DQT87). However, as they and bardment (LHB) (Tsiganis et al., 2005). In each simulation, others (Gaidos, 1995; Adams and Laughlin, 2001) pointed the Sun’s orbit was integrated in the cluster potential to- out, if the Sun remained in this dense environment for long, gether with Jupiter and Saturn and 2200 test particles. The the passing stars could strip the comets away and portions first 2000 test particles were given semimajor axes uni- of the inner OC might not be stable. formly spaced from 4 to 12 AU. The remaining 200 had A comprehensive set of simulations have recently been a ∈ (20, 50) AU and resembles a primordial Kuiper belt. performed by Brasser et al. (2006, hereafter BDL06). Their The stirring of this belt by stellar passages was a measure model assumes that the Sun formed in an embedded star of the damage that close stellar passages did to the system 324 The Solar System Beyond Neptune

Fig. 3. Plot of the relevant timescales for a comet’s evolution as a function of semimajor axis for the models of BDL06. Symbols are the same as in Fig. 1. Three values of td are plotted (near-horizontal solid lines): q = 5 (thin), 7 (medium), and 10 AU (thick). Three 〈ρ〉 3 5 –3 values of tq are plotted (downward-sloping broken lines): = 10 (thin), 10 (medium), and 10 M pc (thick). The orbital period is also plotted (upward-sloping, dotted line).

and is discussed below. The rms values of eccentricity and being ejected. The intersection of the lines of td and tq in- inclination were 0.02 and 0.01 radians respectively. Stellar dicates the value of a at which the lifting is likely to begin encounters were incorporated by directly integrating the and is a measure of the inner edge of the OC. effects of stars passing within a sphere centered on the Sun As it turns out, a mean density of at least 〈ρ〉 ~ 2000 M of radius equal to the Plummer radius for low-density clus- pc–3 is needed to save the comets from Saturn. As can be ters and half a Plummer radius for high-density clusters. The seen from Fig. 3, even when 〈ρ〉 = 105 M pc–3, the comets gravitational influence of the cluster gas was modeled us- with pericenters close to Jupiter can barely be saved, being the tidal force of the cluster potential. For a given solar cause the mean kick in energy from this planet is too strong. orbit, the mean density, 〈ρ〉, was computed by orbit aver- However, for all the densities shown, a fair number of com- aging the density of material encountered. This parameter ets from around Saturn can be saved, with the subsequent proved to be a good measure for predicting the properties OCs being formed ranging in size from a few hundred to of the resulting OC. Many of the results can be understood several thousand AU. by referring to Fig. 3, which is modeled after Fig. 1 and in Figure 4 shows snapshots at the end of five different runs which the symbols have the same meaning. In Fig. 3, the in a–q space. The panels show one run selected from each tidal torquing time from the Plummer potential is computed of the different central densities respectively, with the low- in BDL06 (and is confirmed by numerical integrations) and est density in the bottom panel and the highest in the top the energy diffusion timescales for the pre-LHB planetary left panel. On average 2–18% of the initial sample of com- configuration are numerically obtained as a function of q. ets end up in the OC after 1–3 m.y. A comet is defined to Recall (see, e.g., section 3.7) that once a comet is scat- be part of the OC if it is bound and has q > 35 AU. The tered to semimajor axes a >> q, it tends to diffuse outward models show that the median distance of an object in the 〈ρ〉–1/2 〈ρ〉 –3 at fixed q along the lines of td, until it hits one of two lines. OC scales approximately as when > 10 M pc . If it crosses the period line first, it tends to be ejected. If it The models of BDL06 easily produce objects on orbits crosses the tidal torquing time line first, the comet is usu- like that of (90377) Sedna (Brown et al., 2004) within ally lifted by the tide and stars and can thus be saved from ~1 m.y. in cases where the mean density is 103 M pc–3 or Duncan et al.: Galactic Perturbations of Minor Bodies 325

Fig. 4. Snapshots in a–q space are shown at the end of five different runs from BDL06, one from each set of runs with a different central density. The mean densities the Sun encountered are shown above each panel. The lowest density is at the bottom. Note that the extent and median values of a of the members of the OC increase with decreasing density, as is expected. The positions of (90377)

Sedna, 2000 CR105, and (136199) Eris in order of descending semimajor axis are marked with bullets. Thus it can be seen that objects with orbits like (90377) Sedna and 2000 CR105 only form through this mechanism when the density is high.

higher; one needs mean densities on the order of 104 M without close stellar passages causing disruptive collisions –3 pc to create objects like 2000 CR105 by this mechanism, among bodies in the primordial Kuiper belt down to 20 AU. which are reasonable given the observations of Guthermuth BDL06 concluded that it is therefore improbable that the et al. (2005). Thus the latter object may also be part of the latter object is created by this mechanism. It is possible, OC. although by no means proven, that the Kozai mechanism Close stellar passages can stir the primordial Kuiper belt (coupling between the argument of perihelion, eccentricity, to sufficiently high eccentricities (e > 0.05) (Kenyon and and inclination) associated with mean-motion resonances Bromley, 2002) that collisions become destructive. From the with Neptune may be responsible for raising both the peri- simulations performed it is determined that there is a 50% helion distances and the inclinations of relatively close-in or better chance to stir the primordial Kuiper belt to eccen- objects such as Eris (Gomes et al., 2005; see the chapter tricities e ≥ 0.05 at 50 AU when 〈ρ〉 > 105 M pc–3. Note by Gomes et al.). If so, then the combined simulations of also that in the case of a close stellar encounter that actu- BDL06 and Gomes et al. may provide a rough delineation in ally truncates the belt at 50 AU, there would be a popula- (q–a) space between the inner OC and the region relatively tion of objects on orbits with perihelia like Sedna’s, but unaffected by stellar perturbations. much smaller semimajor axes. This seems inconsistent with the lack of detections of bodies on these orbits. 7.2. The Effect of the Primordial Solar Nebula The orbit of the new object (136199) Eris (Brown et al., 2005) is only reproduced for mean cluster densities on the Drag due to gas in the solar nebula may have been very order of 105 M pc–3, but in the simulations of BDL06 it important in the formation of the OC (de la Fuente Marcos could not come to be on its current orbit by this mechanism and de la Fuente Marcos, 2002; Higuchi et al., 2002). Bras- 326 The Solar System Beyond Neptune

Fig. 5. Snapshots of the endstates in a–q space are shown for five runs from BDL07. All runs have a cluster central density of 103 M pc–3. The headers above each caption are indicative of the parameters used. The horizonal lines correspond to the orbital radius of the planets used. See text for details.

ser et al. (2007, hereafter BDL07) built upon the work of left panel refers to run 1 (disk truncated at 1 and 100 AU, BDL06 by incorporating the aerodynamic drag (adapting using a minimum-mass model: mm1–100), the top-right the results of Adachi et al., 1976) and gravitational potential panel is for run 2 (disk truncated at 6.2 and 10 AU, mini- of the primordial solar nebula. The solar nebula was approx- mum-mass model: mm6–10), the middle-left panel refers imated by the minimum-mass Hayashi model (Hayashi et to run 3 (disk truncated at 6.2 and 100 AU, minimum-mass al., 1985) with scale height 0.047s5/4 AU, where s is the model: mm6–100), the middle-right panel to run 4 (disk cylindrical distance (in AU) from the Sun, and in which the truncated at 6 and 10 AU with Uranus and Neptune present, inner and outer radii were truncated at various distances minimum-mass model: mmun6–10), and run 5 is the bot- from the Sun. In all the simulations, the density of the pri- tom panel (disk truncated at 6.2 and 15 AU with Uranus and mordial solar nebula decayed exponentially with an e-fold- Neptune present, minimum-mass model: mmun6–15). The ing time of 2 m.y. Since the deceleration due to gas drag horizontal lines show the pericenter distances correspond- experienced by a comet is inversely proportional to its size, ing to the orbital distances of Jupiter and Saturn, as well as a typical comet radius of 1.7 km was adopted for most of those of Uranus and Neptune where these are present. the simulations [consistent with the LINEAR observations The panel for run 1 shows that the comets end up either of LPCs of Francis (2005) and using the relation between on circular orbits inside of Jupiter, as co-orbitals of Jupiter mass and absolute magnitude of Weissman (1996)]. The and Saturn or on nearly circular orbits outside of Saturn. numerical simulations of BDL07 followed the evolution of No comets end up in the OC: Close to 60% of the comets comets subject to the gravitational influence of the Sun, end up on circular orbits inside of Jupiter and about 20% Jupiter, Saturn, star cluster, and primordial solar nebula; end up in slightly eccentric orbits at exterior mean-motion some of the simulations included the gravitational influence resonances with Saturn for reasons that are described in of Uranus and Neptune as well. Weidenschilling and Davis (1985). The material that ends Figure 5 shows a vs. q at the end of the simulation for up inside of Jupiter, however, is not in resonance with this runs 1–5 of BDL07, i.e., after 5 m.y. of evolution. The top- planet. Instead, the density of material peaks at a ~ 1.5 AU Duncan et al.: Galactic Perturbations of Minor Bodies 327 and quickly drops to zero at a = 1.8 AU. In other words, are thought subsequently to migrate (see, e.g., Tsiganis et the material is inside the current asteroid belt and in the al., 2005). region where the terrestrial planets are now. Material ends up in this region due to a scattering by Jupiter that places 8. AVENUES TO BE EXPLORED the comets on an orbit with q ~ 2 AU. For this value of q the time to decrease the apocenter away from Jupiter’s or- The models of OC formation described above are still bit is much shorter than the time to receive a strong scat- highly idealized. The “reference model” of section 6 as- tering by Jupiter, so that the orbit quickly circularizes and sumes that the OC formed by the scattering of residual plan- subsequently slowly spirals inward. etesimals by the giant planets in their current configuration For the other runs, in which the inner edge of the disk is with the Sun in its current galactic environment. However, truncated at 6.2 AU, almost no material ends up inside of as noted in that section, there appear to be several difficul- Jupiter. For run 3, in which the outer disk extends to100 AU, ties with this simple model, not the least of which is that it virtually all the material ends up on low-eccentricity orbits does not explain the origins of orbits such as that of (90377) outside Saturn. For runs in which the outer disk is truncated Sedna. Thus, the formation of the OC needs to be studied at much smaller radii (e.g., by photoevaporation) (Hollen- in the context of the Sun’s early galactic environment and bach et al., 2000), the absence of drag near aphelion pro- with a realistic model for planet formation. That is, mod- duces comets on eccentric orbits, often trapped in exterior els must incorporate the fact that the planets were still form- mean-motion resonances. For example, in run 3 (upper right ing during at least the early stages of the formation of the panel of Fig. 5), having the outer edge truncated at 10 AU OC. Planetary migration in the early solar system (Fernán- means that the material does not end up on (nearly) circu- dez and Ip, 1984) appears to have been important in shap- lar orbits outside of Saturn due to gas drag. Instead, the ing the Kuiper belt (Malhotra, 1995; Gomes, 2003; Levison majority of the comets have q interior to 10 AU (78%) with and Morbidelli, 2003; Gomes et al., 2004; Tsiganis et al., almost all these on eccentric orbits with q ~ 9 AU in exterior 2005), and the same is likely true for the OC. Uranus and mean-motion resonances with Saturn. Neptune may even have formed in the Jupiter-Saturn region In general, the simulations of BDL07 show that when (Thommes et al., 1999, 2002), likely changing the fraction the primordial solar nebula extends much beyond Saturn of comets that ended up in the OC (see section 2). Also, as or Neptune, virtually no kilometer-sized comets will end noted in section 7, gas drag may have played a major role up in the inner OC during this phase. Instead, the majority in inner OC formation and collisions may have been im- of the material will be on circular orbits inside of Jupiter if portant in determining which regions of the protoplanetary the inner edge of the disk is well inside Jupiter’s orbit. If disk could populate the OC (Stern and Weissman, 2001; the disk’s inner edge is beyond Jupiter’s orbit, most comets Stern, 2003; but cf. Charnoz and Morbidelli, 2003, 2007). end up on orbits in exterior mean-motion resonances with As noted in section 3.4, it is also quite likely that the OC Saturn when Uranus and Neptune are not present. In those that was formed while the Sun was in its stellar nursery has cases where the outer edge of the disk is close to Saturn or been subjected to pruning and possibly more dramatic shak- Neptune, the fraction of material that ends up in the subse- ing by passage through GMCs and stochastic spiral arms quently formed inner OC is much less than that found in during its multiple circuits around the galactic center over BDL06 for the same cluster densities. the past 4.5 G.y. Future investigations of these effects will All this implies that the presence of the primordial solar doubtless cast further light on this issue. nebula greatly reduces the population of kilometer-sized On the observational side, several groundbased tele- comets to be expected in the inner OC. In order to deter- scopes [e.g., SkyMapper, PanSTARRS and LSST (see mine the effect of the size of the comets on inner OC for- Francis, 2005, for a review)], each capable of studying large mation efficiency, a set of runs with the same initial condi- areas to deeper than 22nd magnitude, may yield several tions but different cometary radii have been performed by hundred new LPC detections over the next few years. Many BDL07. It was determined that the threshold comet size to of these will have perihelia out to and beyond 10 AU and begin producing significant inner OCs is roughly 20 km if will provide critically needed information about the inner the minimum-mass Hayashi model is present for ~2 m.y. OC. In addition, occultation observations (e.g., Chen et al., This implies that the presence of the primordial solar nebula 2007; Roques et al., 2006; Cheng et al., 2006) will provide in the models studied acts as a size-sorting mechanism, with constraints on the TNO population in general while propos- large bodies (such as Sedna) unaffected by the gas drag and als for occultation measurements from space (Lehner, 2006) ending up in the inner OC while kilometer-sized comets offer the tantalizing possibility of detecting comet-sized remain in the planetary region (or in some models in the bodies within the OC itself! Given the dramatic develop- Saturn- or Uranus-Neptune scattered disks). It should be ments of the past few years and the theoretical and observa- noted, however, that comets left on dynamically cold orbits tional prospects over the next few years, it is clear that ex- just beyond the outermost planet are likely to be eventually citing times are ahead in this field. scattered (typically to large semimajor axes in a manner reminiscent of the models described in section 6) in the rel- Acknowledgments. This work was partially supported by the atively gas-free environment in which Uranus and Neptune NASA Planetary Geology and Geophysics Program. M.D. and 328 The Solar System Beyond Neptune

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